Theorem summodclem2 11914

Description: Lemma for summodc 11915. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
summodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5632	. . . . 5 ⊢ (𝑚 = 𝑎 → (ℤ≥‘𝑚) = (ℤ≥‘𝑎))
2	1	sseq2d 3254	. . . 4 ⊢ (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑎)))
3	1	raleqdv 2734	. . . 4 ⊢ (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10689	. . . . 5 ⊢ (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
5	4	breq1d 4093	. . . 4 ⊢ (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
6	2, 3, 5	3anbi123d 1346	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
7	6	cbvrexv 2766	. 2 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8		simplr3 1065	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9		simplr1 1063	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑎))
10		uzssz 9759	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑎) ⊆ ℤ
11	9, 10	sstrdi 3236	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
12		1zzd 9489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
13		simprl 529	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
14	13	nnzd 9584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
15	12, 14	fzfigd 10670	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
16		simprr 531	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
17		f1oeng 6921	. . . . . . . . . . . . . 14 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
18	15, 16, 17	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
19	18	ensymd 6948	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
20		enfii 7049	. . . . . . . . . . . 12 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
21	15, 19, 20	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
22		zfz1iso 11081	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
23	11, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24		isummo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
25		simplll 533	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26		isummo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
27	25, 26	sylan 283	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
28		eleq1w 2290	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
29	28	dcbid 843	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
30		simpr2 1028	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
31	30	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
32		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → 𝑘 ∈ (ℤ≥‘𝑎))
33	29, 31, 32	rspcdva 2912	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → DECID 𝑘 ∈ 𝐴)
34		summodclem2.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
35		eqid 2229	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0))
36		simprll 537	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38		simplr1 1063	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑎))
39		simprlr 538	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
40		simprr 531	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 11913	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
42	41	expr 375	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
43	42	exlimdv 1865	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
44	23, 43	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45		climuni 11825	. . . . . . . . 9 ⊢ ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
46	8, 44, 45	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
47	46	anassrs 400	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48		eqeq2 2239	. . . . . . 7 ⊢ (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦 ↔ 𝑥 = (seq1( + , 𝐺)‘𝑚)))
49	47, 48	syl5ibrcom 157	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
50	49	expimpd 363	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
51	50	exlimdv 1865	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
52	51	rexlimdva 2648	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
53	52	r19.29an 2673	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
54	7, 53	sylan2b 287	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 839   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ⦋csb 3124   ⊆ wss 3197  ifcif 3602   class class class wbr 4083   ↦ cmpt 4145  –1-1-onto→wf1o 5320  ‘cfv 5321   Isom wiso 5322  (class class class)co 6010   ≈ cen 6898  Fincfn 6900  ℂcc 8013  0cc0 8015  1c1 8016   + caddc 8018   < clt 8197   ≤ cle 8198  ℕcn 9126  ℤcz 9462  ℤ_≥cuz 9738  ...cfz 10221  seqcseq 10686  ♯chash 11014   ⇝ cli 11810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811

This theorem is referenced by:  summodc  11915